A law of the iterated logarithm for an estimate of frequency

On the law of the iterated logarithm.

The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution...

An Analogue of the Law of the Iterated Logarithm

the innovation of a statistical model to estimate dependable rainfall (dr) and develop it for determination and classification of drought and wet years of iran

آب حاصل از بارش منبع تأمین نیازهای بی شمار جانداران به ویژه انسان است و هرگونه کاهش در کم و کیف آن مستقیماً حیات موجودات زنده را تحت تأثیر منفی قرار می دهد. نوسان سال به سال بارش از ویژگی های اساسی و بسیار مهم بارش های سالانه ایران محسوب می شود که آثار زیان بار آن در تمام عرصه های اقتصادی، اجتماعی و حتی سیاسی- امنیتی به نحوی منعکس می شود. چون میزان آب ناشی از بارش یکی از مولفه های اصلی برنامه ...

A Law of the Iterated Logarithm for General Lacunary Series

This was first proved for Bernoulli random variables by Khintchine. Salem and Zygmund [SZ2] considered the case when the Xk are replaced by functions ak cosnkx on [−π, π] and gave an upper bound ( ≤ 1) result; this was extended to the full upper and lower bound by Erdös and Gál [EG]. Takahashi [T1] extends the result of Salem and Zygmund: Consider a real measurable function f satisfying f(x + 1...

A Law of the Iterated Logarithm for Arithmetic Functions

Let X,X1, X2, . . . be a sequence of centered iid random variables. Let f(n) be a strongly additive arithmetic function such that ∑ p<n f2(p) p → ∞ and put An = ∑ p<n f(p) p . If EX2 < ∞ and f satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: lim sup N→∞ ∑N n=1 f(n)Xn AN √ 2N log logN a.s. = ‖X‖2. We also prove the validity of the corresponding weighted...